bumblebee~ What a cool book!!!! I think it's wonderful you found some of your great grandmother's patterns. That is so special. I wish I had my grandmother's patterns.
bhazell~ Wow...you are hookin right along!!! Way to go!
Looks like you started reducing too late. As you move out from your original circle the amount of additional stitches required for a doily are less for a beenie. If the hat doesn't curve nicely then the pattern is calling for too many additional stitches in each round. At the end it looks like the additional stitches were stopped all at once giving you a suddenly straight edge. Like you might use for a basket. You could always start reducing at this stage and use it as a beret. Although the sides are already too long for a beret.
I would rip out to about the 4th row and use a few less stitches in each succeeding row. That will correct the problem. Once you've done a few of these you won't even have to follow a pattern you'll know, by the curve of the hat, as your working what to add to get a nice form fit.
greensfamilyfarm~ That is exactly what I did the first time I tried to do a beanie cap. I'll send you a pattern that jen4 shared w/ me that I found so much easier for me to read and they make adorable beanies.